173 lines
2.8 KiB
Plaintext
173 lines
2.8 KiB
Plaintext
# Differenciation
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=> integration.gmi Also see Integration
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Differenciation was discovered/pioneered by Isaac Newton
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It allows you to find the gradient equation for a curve (eg x²+3x+6)
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You can easily find the gradient between two points on a curve by using the equation
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```
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change in y / change in x
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```
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But if you want to find the gradient at a specific point on a graph, you have to draw a tangent... Or DO you?
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Well, if you pick two points on a graph (f(x): y = x²+3x+6) where
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```x = 1```
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and
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```x = 1 + h```
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Where h is a number that is very close to 0.
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You can find the gradient between these two points:
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```
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change in y
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-----------
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change in x
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=>
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((1+h)² + 3(1+h) + 6) - (1² + 3(1) + 6)
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---------------------------------------
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(1 + h) - 1
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=>
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( h² + 5h + 10) - (10)
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----------------------
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h
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=>
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h² + 5h
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-------
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h
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=>
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h + 5
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=>
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5 (because h is nearly 0)
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```
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...thus the gradient at point x=1 is 5
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This proof is called First Principle and can be written:
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```
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f(x+h) - f(x)
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-------------
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f'(x) = lim h
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h→0
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```
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This takes a while so a quicker way we can do this is by differenciating the curve equation
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We do this by multiplying the coefficient by the number that the x is a power to and then minusing one from the the power:
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```
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f(x) = x² + 3x + 6
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f'(x) = 2x + 3
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dy
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-- = 2x + 3
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dx
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```
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f'(x) and dy/dx are just different ways of denoting a differenciated equation
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Then we can simply substitute 1 into the equation:
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```
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2(1) + 3 = 5
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```
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And boom, that is differenciation; finding the gradient line of a curve
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If you draw the gradient curve then you will see that the point it crosses the x axis, where x=0, the gradient=0 as that is the turning point of the quadratic equation:
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```
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0 = 2x + 3
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-3 = 2x
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-1.5 = x
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```
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And so x = -1.5 is the minimum point for x
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```
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x² + 3x + 6:
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| | |
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| || (1. Sides of the V shape are getting steeper)
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' '|
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\./ | (2. Turning point is -1.5)
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_______________|________________
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2x + 3:
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(2. Crosses |/ (1. Because the line is getting higher)
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the x axis /
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at -1.5) /|
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/ |
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____________/__|________________
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/ |
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/ |
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```
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Additionally, if an equation has roots or fractions, you can use the index laws to put the equation in indicies form:
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=> index-laws.gmi Index Laws
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```
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x³ + sqrt(x)
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= x³ + x^(1/2)
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2x² + 1/x
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= 2x² + x^(-1)
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```
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Questions:
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1.
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a) Differenciate x³ - 4x² + 6x - 1
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b) Find the gradient at x=3
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=> integration.gmi Also see Integration
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for finding the area under a curve, the reverse process of differenciation
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Answers
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↓↓↓↓↓↓↓
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Answers:
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1.
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a) 3x² - 8x + 6
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b) 9
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